hermitian

Given such a from on a complex vector space, it defines a natural way to associate the space with its dual. Knowing this, we call the dual map to a map on the space its hermitian adjoint. If a map is equal to its hermitian adjoint, it is said to be hermitian.

A hermitian operator is one that satisfies &lt Hx|y &gt = &lt x|Hy &gt, where x and y are real or complex vectors or functions and H is a linear operator and &lt x|y &gt denotes the inner product of x and y.

Some properties: Suppose we know that Hx = λx where &lambda is an eigenvalue of the hermitian operator H. Then we can prove that &lambda must be real. Consider:

Given that x and y are eigenvectors of the hermitian operator H, and &lambda and ɸ are the eigenvalues of H to x and y respectively (note also that &lambda ≠ ɸ), then we can prove that x and y are orthogonal.